Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#2950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C23×C11 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C11×#117 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,14)(16,17)(18,26)(19,23)(20,32)(21,24)(22,28)(25,34)(27,40)(29,39)(30,38)(31,44)(33,36)(35,42)(37,43)(41,47)(45,46);(16,18,21,29)(17,22,24,36)(19,27,30,41)(20,33,34,28)(23,35,38,46)(25,39,32,26)(31,45,43,42)(37,47,44,40);(12,14,15,13)(16,19,21,30)(17,23,24,38)(18,27,29,41)(20,31,34,43)(22,35,36,46)(25,37,32,44)(26,40,39,47)(28,42,33,45);(17,24)(19,30)(22,36)(25,32)(26,39)(27,41)(31,43)(42,45)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1682);